Height representation of XOR-Ising loops via bipartite dimers C - - PowerPoint PPT Presentation

Height representation of XOR-Ising loops via bipartite dimers

C´ edric Boutillier (UPMC) B´ eatrice de Tili` ere (UPMC) LAGA Universit´ e Paris Nord – March 11, 2015

The Ising model and the XOR-Ising model

The Ising model

◮ Let G = (V, E) be a finite graph embedded in the plane ◮ spin configuration σ : V −

→ {−1, +1}

◮ σ assigns to every vertex x a spin σx ∈ {−, +}

+1/−1 are represented by green/blue dots.

The Ising model

◮ Edges of G are assigned positive coupling constants:

J = (Je)e∈E.

◮ Ising Boltzmann measure:

∀ σ ∈ {−1, 1}V, PIsing(σ) = 1 ZIsing(G, J) exp  

e=xy∈E

Jxyσxσy   , where ZIsing(G, J) =

• σ∈{−1,1}V

exp  

e=xy∈E

Jxyσxσy   is the Ising partition function.

The XOR-Ising model

= = = = × × × ×

σ σ′ ξ = σσ′

Ising model on G, J = (Je)e∈E Ising model on G, J = (Je)e∈E XOR-Ising model on G, J = (Je)e∈E

The XOR-Ising model

= = = = × × × ×

σ σ′ ξ = σσ′

XOR-Ising model on G, J = (Je)e∈E Ising model on G, J = (Je)e∈E Ising model on G, J = (Je)e∈E

Conjecture for the XOR-Ising model

Conjecture (Wilson (11), Ikhlef–Picco–Santachiara) The scaling limit of polygon configurations separating ±1 clusters

• f the critical XOR-Ising model are contour lines of the Gaussian

free field, with the heights of the contours spaced √ 2 times as far apart as they are for [...] the double dimer model on the square lattice.

Result

Theorem (B–dT)

◮ Polygon configurations of the XOR-Ising model have the same

law as a family of contours in a bipartite dimer model.

◮ This family of contours are the level lines of a restriction of

the height function of this bipartite dimer model. Remark Proved when the graph G is embedded in a surface of genus g, or when G is planar, infinite.

◮ When the XOR-Ising model is critical, so is the bipartite

dimer model.

◮ Using results of [dT] on the convergence of the height

function, this gives partial proof of Wilson’s conjecture.

Contour expansion of the Ising partition function [Kramers & Wannier]

Low temperature expansion

◮ Polygon configuration: subset of edges s.t. each vertex is

incident to an even number of edges.

◮ Write

eJeσxσy = eJe(δ{σx=σy} + e−2Jeδ{σx=σy}). The partition function is then equal to (LTE): ZIsing(G, J) =

• σ∈{−1,1}V
• e=xy∈E

eJeσxσy = C

• P∗∈P(G∗)
• e∗∈P∗

e−2Je.

◮ Geometric interp: polygon config. separate clusters of ±1

spins.

High temperature expansion

◮ Write,

eJeσxσy = cosh(Je)(1 + σxσy tanh(Je)). The partition function is then equal to (HTE): ZIsing(G, J) =

• σ∈{−1,1}V
• e=xy∈E

eJeσxσy = C′

P∈P(G)

• e∈P

tanh(Je).

◮ No geometric interpretation using spin configurations.

Mixed contour expansion for the double Ising model

The double Ising model

◮ Take 2 independent copies (red/blue) of an Ising model on G,

with coupling constants J.

◮ Using the LTE, consider the probability measure P2-Ising:

if P∗, P∗ are two polygon configurations. P2-Ising(P∗, P∗) = C2

e∗∈P∗ e−2Je e∗∈P∗ e−2Je

Z2-Ising(G, J) , where Z2-Ising(G, J) = ZIsing(G, J)2.

The double Ising model

◮ Let P∗, P∗ be two polygon configurations. ◮ Consider the superimposition P∗ ∪ P∗. ◮ Define two new edge configurations:

◮ Mono(P∗, P∗): monochromatic edges. ◮ Bi(P∗, P∗): bichromatic edges.

Monochromatic edges

Monochromatic edge configuration of P∗ ∪ P∗

Lemma Mono(P∗, P∗) is the polygon configuration separating ±1 clusters

• f the corresponding XOR-Ising spin configuration.

Goal: understand the law of monochromatic edge configurations.

Bichromatic edge configurations

◮ Let (P∗, P∗) be two polygon configurations. ◮ Mono(P∗, P∗) splits the surface into connected comp. (Σi)i.

Σ6 Σ7 Σ2 Σ5 Σ9 Σ8 Σ4 Σ1 Σ3

Lemma For every i, the restriction of Bi(P∗, P∗) to Σi is the LTE of an Ising configuration on GΣi, with coupling constants (2Je).

Probability of monochromatic configurations

Lemma Let P∗ be a polygon configuration, separating the surface into n connected components. For every i, let P∗

i be a polygon

configuration of G∗

Σi.

Then, there are 2n pairs of polygon configurations (P∗, P∗) having P∗ as monochromatic edges, and P∗

1, · · · , P∗ n as bichromatic edges.

Denote by W(P∗) the contribution to Z2-Ising(G, J) of the pairs of polygon configurations (P∗, P∗) such that Mono(P∗, P∗) = P∗. Corollary

◮ W(P∗) = C

• e∗∈P∗ e−2Je n

i=1

• 2ZLT(G∗

Σi, 2J)

• ◮ Z2-Ising(G, J) =

P∗∈P(G∗) W(P∗)

P2-Ising(Mono = P∗) =

W(P∗) Z2-Ising(G,J).

Mixed contour expansion

W(P∗) = C

• e∗∈P∗ e−2Je n

i=1

• 2ZLT(G∗

Σi, 2J)

• .

Idea [Nienhuis]: high temperature expansion in each connected component Σi. ZLT(G∗

Σi, 2J) = C(Σi)ZHT(GΣi, 2J).

Low temp. expansion on G∗

Σi

High temp. expansion on GΣi.

Mixed contour expansion

Proposition For every polygon configuration P∗,

W(P∗) = C

• e∗∈P∗

2e−2Je 1 + e−4Je

• {P∈P(G): P∗∩P=∅}
• e∈P

1 − e−4Je 1 + e−4Je

• P2-Ising(Mono = P∗) =
• e∗∈P∗
• 2e−2Je

1+e−4Je

• {P∈P(G): P∗∩P=∅}
• e∈P
• 1−e−4Je

1+e−4Je

• P∗∈P(G∗)

···

Higher genus

If the graph is embedded in a surface Σ of genus g ≥ 0.

◮ Consider H1(Σ, Z/2Z) ≃ {0, 1}2g. ◮ Family of Ising models, indexed by ε ∈ {0, 1}2g. ◮ The double Ising model partition function is defined as:

Z2-Ising(G, J) =

• ε∈{0,1}2g

Zε

Ising(G, J)2.

From mixed polygon configurations to dimers

The graph GQ = (VQ, EQ)

The dimer model on GQ

dimer configuration of GQ: a subset of edges M such that each vertex is incident to exactly on edge of M

The dimer model on GQ

dimer configuration of GQ: a subset of edges M such that each vertex is incident to exactly on edge of M

The dimer model on GQ

dimer configuration of GQ: a subset of edges M such that each vertex is incident to exactly on edge of M weight function ν on the edges Dimer Boltzmann measure: Pdimer(M) ∝

e∈EQ νe

First step: from polygons to 6-vertex [Nienhuis]

1 2 3 4 5 6 1 2 3 4 5 6

Local mapping

Weights: ω12 =

2e−2Je 1+e−4Je , ω34 = 1−e−4Je 1+e−4Je , ω56 = 1.

First step: from polygons to 6-vertex [Nienhuis]

1 2 3 4 5 6 1 2 3 4 5 6

Local mapping

Weights: ω12 =

2e−2Je 1+e−4Je , ω34 = 1−e−4Je 1+e−4Je , ω56 = 1.

Second step: from 6V to dimers [Wu-Lin, Dub´ edat]

1 2 3 4 5 6 = 1 +

2 2

1 1 1 1 1 ω ω ω 12

34 12

ω34 ω ω ω ω 1 1

12 12 34 34

ω 12 ω 12 ω 34 ω 34

34

ω12 ω

Local mapping

Conclusion

◮ To every dimer configuration M of GQ, assign

Poly(M) = (Poly1(M), Poly2(M)), the pair of polygon configurations given by the mappings. Theorem For every polygon configuration P∗ of G∗, P2-Ising(Mono = P∗) = Pdimer(Poly1 = P∗)

Height function for bipartite dimers (Thurston)

Height function for bipartite dimers (Thurston)

1 1 1 −1 1 1 1 1 1 1 1 1 1 1 1 −1 1 1 1 −1 −1 −1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 −1

Height function for bipartite dimers (Thurston)

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

Height function for bipartite dimers (Thurston)

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

The critical XOR-Ising model on isoradial graphs

Isoradial graphs

A graph G is isoradial if it is planar and can be embedded in the plane in such a way that all faces are inscribed in a circle of radius 1, and that the circumcenters are in the interior of the faces. [Duffin, Mercat, Kenyon]

Isoradial graphs

Isoradial graphs

Isoradial graphs

Isoradial graphs

Associated rhombus graph

Critical Ising model on isoradial graphs

◮ To each edge e is naturally associated an angle θe ◮ The Ising model defined on an isoradial graph G is

critical if the coupling constants are given by: Je = 1 2 log 1 + sin θe cos θe

• .

(Z-invariance + duality [Baxter], proof in period. case [Li, Duminil–Cimasoni])

Example: G = Z2: θe = π

4 , Je = 1 2 log(1 +

√ 2).

◮ The corresponding bipartite graph GQ is also isoradial,

and the weights are the critical dimer weights:

θe e

sin sin θ cosθ θ cos θ

θ

1 1 1 1

Back to Wilson’s conjecture

Conjecture (Wilson) The scaling limit of polygon configurations separating ±1 clusters

• f the critical XOR-Ising model are contour lines of the Gaussian

free field, with the heights of the contours spaced √ 2 times as far apart as they are for [...] the double dimer model on the square lattice. Theorem (B–dT) XOR-polygon configurations of the double Ising model on G have the same law as level lines of a restriction of the height function of the bipartite dimer model on GQ, with an explicit coupling. Theorem (dT) The height function (as a random distribution) of the critical dimer model defined on a bipartite graph converges weakly in law to

1 √π

a Gaussian free field of the plane.

Back to Wilson’s conjecture

Suppose we had strong form of convergence, allowing for convergence of level lines. Then: level lines of hε → level lines of GFF (k, k ∈ Z) (√πk, k ∈ Z) (k + 1

2, k ∈ Z)

(

√π 2 (2k + 1), k ∈ Z)

XOR loops For the critical double dimer model. The height function is hε

1 − hε 2, where h1 and h2 are independent, and each converges

weakly in distribution to

1 √π a Gaussian free field. Thus, h1 − h2

converges weakly in distribution to

√ 2 √π a Gaussian free field.

level lines of hε

1 − hε 2

→ level lines of GFF (k, k ∈ Z) (

√π √ 2 k, k ∈ Z)

(k + 1

2, k ∈ Z)

(

√π 2 √ 2(2k + 1), k ∈ Z)

d-dimer loops